It is basically for when a player is coaching another player ... if you make a move which is a mistake then you can "take back" the move and move something else
Take back game


A bit more detail here in the FAQ...
http://support.chess.com/index.php?_m=knowledgebase&_a=viewarticle&kbarticleid=22&nav=0,6
Could I direct your attention towards previous topics (e.g. http://www.chess.com/forum/view/general/taking-back-a-move) where ideas about undoing or cancelling a move have been discussed, including the logistics, and the pros and cons.
The facility of TAKING BACK has now been introduced within another form of unrated game. From Start New Game, under Show more advanced options, the Rated option offers a dropdown menu now containing THREE items: Rated game, Unrated game and Takeback game.
Within a TAKEBACK game, an extra button "TAKEBACK" is displayed, alongside SUBMIT MOVE, OFFER DRAW, CONDITIONAL MOVES. Both sides can takeback at any time. When it is your move, clicking on TAKEBACK undoes your opponent's previous move - and it becomes their turn again. When it is NOT your move, clicking on TAKEBACK undoes your previous move - and it becomes your turn again. This can be repeated all the way back to the beginning of the game. You can't undo CHECKMATE (or once the game is over). http://www.chess.com/forum/view/community/new-features--fixes---jan-14-2008
I noticed today next to a "rated" and "unrated" game, that I could select a "take back game" when creating an open challenge.
I didn't know what this is, and the help/FAQ/etc didn't tell me.
Who knows what this take back game option is about?